HAL CCSD
Transgressions of the Euler class and Eisenstein cohomology of GLN(Z)
Bergeron, Nicolas
Charollois, Pierre
Garcia, Luis
Département de Mathématiques et Applications - ENS Paris (DMA) ; École normale supérieure - Paris (ENS-PSL) ; Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)
Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)) ; Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)
Department of Mathematics (UCL London) ; University College of London [London] (UCL)
International audience
ISSN: 0289-2316
EISSN: 1861-3624
Japanese Journal of Mathematics
Springer Verlag
hal-02886362
https://ens.hal.science/hal-02886362
https://ens.hal.science/hal-02886362/document
https://ens.hal.science/hal-02886362/file/Takagi-final-JJM1822.pdf
https://ens.hal.science/hal-02886362
Japanese Journal of Mathematics, 2020, 15 (2), pp.311-379. ⟨10.1007/s11537-019-1822-6⟩
DOI: 10.1007/s11537-019-1822-6
info:eu-repo/semantics/altIdentifier/doi/10.1007/s11537-019-1822-6
en
[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT]
[MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT]
info:eu-repo/semantics/article
Journal articles
These notes were written to be distributed to the audience of the first author's Takagi lectures delivered June 23, 2018. These are based on a work-in-progress that is part of a collaborative project that also involves Akshay Venkatesh. In this work-in-progress we give a new construction of some Eisenstein classes for GL N (Z) that were first considered by Nori [41] and Sczech [44]. The starting point of this construction is a theorem of Sullivan on the vanishing of the Euler class of SL N (Z) vector bundles and the explicit transgression of this Euler class by Bismut and Cheeger. Their proof indeed produces a universal form that can be thought of as a kernel for a regularized theta lift for the reductive dual pair (GL N , GL 1). This suggests looking to reductive dual pairs (GL N , GL k) with k ≥ 1 for possible generalizations of the Eisenstein cocycle. This leads to fascinating lifts that relate the geometry/topology world of real arithmetic locally symmetric spaces to the arithmetic world of modular forms. In these notes we don't deal with the most general cases and put a lot of emphasis on various examples that are often classical.
2020-09
info:eu-repo/semantics/OpenAccess